Compact-support conjecture for self-similar states

Let $(A,L)$ satisfy the requirements of the strictly contractive dual iterated function system, let $\phi_0\in\mathcal{S}_{\mathrm{c}}(A)$, and let $\phi_\pi$ be the self-similar state associated with $\pi\in\mathring{\Delta}^k$. Compact-support conjecture. If $(A,L)$ is locally semi-consistent, then $\phi_\pi$ is compactly supported for every $\pi\in\mathring{\Delta}^k$. This is a proposed support property for self-similar states in the strictly contractive noncommutative setting, motivated by the compact attractor of the associated classical dual iterated function system; it remains open in the supplied source.